When using aerial imagery to trace detail it is often useful to "overzoom" past the highest resolution they offer in order to place points more precisely etc. Some sources such as Bing simply return an error message when they don't have a zoom level allowing the previous (now blurry) zoom level to stay on the screen. Others, such as Mapbox and MapQuest show row after row, column after column of the same useless placeholder image.

Is there a setting or Plugin for JOSM that allows me to specify a image that should be ignored and never rendered in the background?

A workaround: you can specify/edit the max zoom level of each imagery source in your imagery preferences: the number in [ ]. I guess you will need to re-add (to your layers list) a imagery layer to activate this setting, in case you already had bing in your current layers.

Another workaround: once the highest-resolution zoom level you need is loaded, if you right-click the layer in the Layers pane and uncheck "Auto zoom" it will overzoom beyond that zoom level instead of trying to request any other tiles.